@Markiyan Hirnyk   you're right! Cases  a=0  or  h=0  or  k=0  require special consideration.

@John 2014  Your problem is reduced to solving a system of linear equations (see the above link).

Solution for the base L1 (the remaining items can be solved similarly):

Poly:=x*(t^2+1)+y*(t-2)+z*(t+3);  Poly2:=a1*t^2+ a2*t+ a3;

Sys:={seq(coeff(Poly,t,n)=coeff(Poly2,t,n), n=0..2)}; #  The system of linear equations

solve(Sys, {x,y,z});

1) Use Maple syntax for imputs of formulas, for example,  a1*t^2+a2*t+a3 . Your formulas are incomprehensible.

2) See  the question

• Showing that a list is an ordered basis for a subspace

@mahmood180   Replace  Int  by  int

@Markiyan Hirnyk 

execution.mws

Carl's code is working:

Sol:=fsolve({F,Fw,Fk,Ft}, map(`=`,{w,k,ki,T},0..infinity));
evalf(eval({F,Fw,Fk,Ft}, Sol));

@apm   Since  limit  is weaker than  MultiSeries:-limit  then before  limit  you have to use a simplifying transformation as shown by Axel Vogt. In addition, instead of  combine, you can use  expand  command:

restart;

Expr:=sqrt(-2*cos(alpha)*cos(alpha+d)+2-2*sin(alpha+d)*sin(alpha))/d:

limit(expand(Expr), d = 0, left);  limit(expand(Expr), d = 0, right);

                                                         -1

                                                          1

@Markiyan Hirnyk  Ill-conditioned and ill-posed systems of linear algebraic equations are the different things.

@Carl Love 

1) Thanks for comment on zero entries.

2)  Transpose of M does not help.  

In Maple 12:

 convert(M^%T, list)[];

@Carl Love  Done in Maple 12 Classic.

# Return the angle of a point in interval [0, 2*Pi).

Angle:= proc(P::[realcons,realcons])

local arg:= evalf(arctan(P[2],P[1]));

     `if`(arg < 0, evalf(arg+2*Pi), arg)

end proc:

SortByAngle:= (A::Matrix)->

     Matrix(sort(convert(A^%T, listlist), (P,Q)-> Angle(P) < Angle(Q)))^%T:

M:= < -1,2,3,1,1,-1 ; 0,2,3,1,0,0 >:

SortByAngle(M);

@wingjammer   My example is of a general nature, you can write in the brackets the order of columns that you need.

Specify your problem more clearly. I do not understand what you want in the general case.

@Carl Love   Your code is compact and elegant, but conceptually more complex and less effective:

t := time():  SpM(1000):  time()-t;

t := time():  Am(1000):  time()-t;

Of course, this can be done in different ways. Here are 2 other ways:

restart;

f:=x+y+z:

V:=<seq(diff(f, var), var=[x,y,z])>;

restart;

f:=x+y+z:  V:=Vector():  var:=[x,y,z]:

for i to nops(var) do

V(i):=diff(f, var[i]):

od:

V;

@Carl Love   It is sufficient to find the exact roots of the polynomial (the first command in my answer)

Another example:

factor(3*x^2-7);

factor(3*x^2-7, {sqrt(3), sqrt(7)});

 -op(1,%)*3*``(expand(op(2,%)/sqrt(3)))*``(-expand(op(3,%)/sqrt(3)));

@Carl Love   In your example we can do exact symbolic factorization:

A := factor(x^4+1, sqrt(2));

B := ``(op(1, A)*op(2, A))*op(3, A);